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        Trilateration三边测量定位算法
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        <h2>
            Trilateration三边测量定位算法
        </h2>

        <div class="body-main-content">
            <h2 id="Trilateration（三边测量）是一种常用的定位算法"><a href="#Trilateration（三边测量）是一种常用的定位算法" class="headerlink" title="Trilateration（三边测量）是一种常用的定位算法"></a>Trilateration（三边测量）是一种常用的定位算法</h2><span id="more"></span>

<blockquote>
<p>已知三点位置 (x1, y1), (x2, y2), (x3, y3)<br>已知未知点 (x0, y0) 到三点距离 d1, d2, d3<br>以 d1, d2, d3 为半径作三个圆，根据毕达哥拉斯定理，得出交点即未知点的位置计算公式：<br>( x1 - x0 )2 + ( y1 - y0 )2 = d12<br>( x2 - x0 )2 + ( y2 - y0 )2 = d22<br>( x3 - x0 )2 + ( y3 - y0 )2 = d32</p>
</blockquote>
<p><img src="" data-original="https://img-blog.csdnimg.cn/20190923114350915.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3UwMTM2MDUwNjA=,size_16,color_FFFFFF,t_70" alt="ar5hNR.jpg"></p>
<blockquote>
<p>设未知点位置为 (x, y)， 令其中的第一个球形 P1 的球心坐标为 (0, 0)，P2 处于相同纵坐标，球心坐标为 (d, 0)，P3 球心坐标为 (i, j)，三个球形半径分别为 r1, r2, r3，z为三球形相交点与水平面高度。则有：</p>
</blockquote>
<blockquote>
<p>r12 = x2 + y2 + z2<br>r22 = (x - d)2 + y2 + z2<br>r32 = (x - i)2 + (y - j)2 + z2<br>当 z = 0 时， 即为三个圆在水平面上相交为一点，首先解出 x:</p>
</blockquote>
<blockquote>
<p>x = (r12 - r22 + d2) / 2d<br>将公式二变形，将公式一的 z2 代入公式二，再代入公式三得到 y 的计算公式：</p>
</blockquote>
<blockquote>
<p>y = (r12 - r32 - x2 + (x - i)2 + j2) / 2j</p>
</blockquote>
<h2 id="JAVA算法实现"><a href="#JAVA算法实现" class="headerlink" title="JAVA算法实现"></a>JAVA算法实现</h2><figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br></pre></td><td class="code"><pre><span class="line">public class Algorithem &#123;</span><br><span class="line">    public static void main(String[] args)&#123;</span><br><span class="line">        double[] xy = Algorithem.trilateration(12641371.971, 4138703.5211, 6, 12641381.9026, 4138706.4714, 6, 12641370.7839, 4138708.7705, 6);</span><br><span class="line">        System.out.println(xy[0]+&quot;::&quot;+xy[1]);</span><br><span class="line">    &#125;</span><br><span class="line">    public static double[] trilateration(double x1,double y1,double d1, double x2, double y2,double d2, double x3, double y3, double d3)</span><br><span class="line">    &#123;</span><br><span class="line">        double []d=&#123;0.0,0.0&#125;;</span><br><span class="line">        double a11 = 2*(x1-x3);</span><br><span class="line">        double a12 = 2*(y1-y3);</span><br><span class="line">        double b1 = Math.pow(x1,2)-Math.pow(x3,2) +Math.pow(y1,2)-Math.pow(y3,2) +Math.pow(d3,2)-Math.pow(d1,2);</span><br><span class="line">        double a21 = 2*(x2-x3);</span><br><span class="line">        double a22 = 2*(y2-y3);</span><br><span class="line">        double b2 = Math.pow(x2,2)-Math.pow(x3,2) +Math.pow(y2,2)-Math.pow(y3,2) +Math.pow(d3,2)-Math.pow(d2,2);</span><br><span class="line">        d[0]=(b1*a22-a12*b2)/(a11*a22-a12*a21);</span><br><span class="line">        d[1]=(a11*b2-b1*a21)/(a11*a22-a12*a21);</span><br><span class="line">        return d;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h2 id="D3-js实现"><a href="#D3-js实现" class="headerlink" title="D3.js实现"></a>D3.js实现</h2><p>定义三个圆的坐标及半径，计算出交点的坐标 (obj_x, obj_y).</p>
<figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br></pre></td><td class="code"><pre><span class="line">var x_0 = 150, y_0 = 150;</span><br><span class="line">var x_1 = x_0, y_1 = y_0, d = 150, x_2 = x_0 + d, x_3 = 225, y_3 = 315, r = 100;</span><br><span class="line">var i = x_3 - x_0, j = y_3 - y_0;</span><br><span class="line">var x = (Math.pow(r, 2) - Math.pow(r, 2) + Math.pow(d, 2)) / (2 * d) + x_0;</span><br><span class="line">var obj_x = x + x_0;</span><br><span class="line">var y = (Math.pow(r, 2) - Math.pow(r, 2) - Math.pow(x, 2) + Math.pow((x - i), 2)</span><br><span class="line">    + Math.pow(j, 2)) / (2 * j);</span><br><span class="line">var obj_y = y + y_0;</span><br></pre></td></tr></table></figure>

<h2 id="绘出圆形及交点："><a href="#绘出圆形及交点：" class="headerlink" title="绘出圆形及交点："></a>绘出圆形及交点：</h2><figure class="highlight plaintext"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br></pre></td><td class="code"><pre><span class="line">svg.append(&quot;circle&quot;).attr(&quot;cx&quot;, x_1)</span><br><span class="line">    .attr(&quot;cy&quot;, y_1).attr(&quot;r&quot;, r)</span><br><span class="line">    .style(&quot;fill&quot;, &quot;blue&quot;).style(&quot;opacity&quot;, 0.3);</span><br><span class="line">svg.append(&quot;circle&quot;).attr(&quot;cx&quot;, x_2)</span><br><span class="line">    .attr(&quot;cy&quot;, y_0).attr(&quot;r&quot;, r)</span><br><span class="line">    .style(&quot;fill&quot;, &quot;red&quot;).style(&quot;opacity&quot;, 0.4);</span><br><span class="line">svg.append(&quot;circle&quot;).attr(&quot;cx&quot;, x_3)</span><br><span class="line">    .attr(&quot;cy&quot;, y_3).style(&quot;opacity&quot;, 0.5)</span><br><span class="line">    .attr(&quot;r&quot;, r).style(&quot;fill&quot;, &quot;yellow&quot;);</span><br><span class="line">svg.append(&quot;circle&quot;).attr(&quot;cx&quot;, obj_x)</span><br><span class="line">    .attr(&quot;cy&quot;, obj_y).attr(&quot;r&quot;, 3)</span><br><span class="line">    .style(&quot;fill&quot;, &quot;red&quot;);</span><br></pre></td></tr></table></figure>

<h2 id="多于用实现定位算法，与之相同的也许还可以采用K-Means或Overlap聚类算法，后两者在地图标注上使用比较广。"><a href="#多于用实现定位算法，与之相同的也许还可以采用K-Means或Overlap聚类算法，后两者在地图标注上使用比较广。" class="headerlink" title="多于用实现定位算法，与之相同的也许还可以采用K-Means或Overlap聚类算法，后两者在地图标注上使用比较广。"></a>多于用实现定位算法，与之相同的也许还可以采用K-Means或Overlap聚类算法，后两者在地图标注上使用比较广。</h2>
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        <ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#Trilateration%EF%BC%88%E4%B8%89%E8%BE%B9%E6%B5%8B%E9%87%8F%EF%BC%89%E6%98%AF%E4%B8%80%E7%A7%8D%E5%B8%B8%E7%94%A8%E7%9A%84%E5%AE%9A%E4%BD%8D%E7%AE%97%E6%B3%95"><span class="toc-text">Trilateration（三边测量）是一种常用的定位算法</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#JAVA%E7%AE%97%E6%B3%95%E5%AE%9E%E7%8E%B0"><span class="toc-text">JAVA算法实现</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#D3-js%E5%AE%9E%E7%8E%B0"><span class="toc-text">D3.js实现</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%BB%98%E5%87%BA%E5%9C%86%E5%BD%A2%E5%8F%8A%E4%BA%A4%E7%82%B9%EF%BC%9A"><span class="toc-text">绘出圆形及交点：</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%A4%9A%E4%BA%8E%E7%94%A8%E5%AE%9E%E7%8E%B0%E5%AE%9A%E4%BD%8D%E7%AE%97%E6%B3%95%EF%BC%8C%E4%B8%8E%E4%B9%8B%E7%9B%B8%E5%90%8C%E7%9A%84%E4%B9%9F%E8%AE%B8%E8%BF%98%E5%8F%AF%E4%BB%A5%E9%87%87%E7%94%A8K-Means%E6%88%96Overlap%E8%81%9A%E7%B1%BB%E7%AE%97%E6%B3%95%EF%BC%8C%E5%90%8E%E4%B8%A4%E8%80%85%E5%9C%A8%E5%9C%B0%E5%9B%BE%E6%A0%87%E6%B3%A8%E4%B8%8A%E4%BD%BF%E7%94%A8%E6%AF%94%E8%BE%83%E5%B9%BF%E3%80%82"><span class="toc-text">多于用实现定位算法，与之相同的也许还可以采用K-Means或Overlap聚类算法，后两者在地图标注上使用比较广。</span></a></li></ol>
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    for (let i = 0; i <= nPts; i++) { // add paths + animate
        const p = document.createElementNS("http://www.w3.org/2000/svg", "path")

        gsap.set(p, {
            x: 250,
            y: 250,
            attr: {
                class: 'line' + i,
                d: 'M' + pts[i] + ' C' + gsap.utils.wrap(pts, [i + 2]) + ' ' + gsap.utils.wrap(pts, [i - 5]) +
                    ' ' + gsap.utils.wrap(pts, [i - 2]),
                // d:'M'+pts[i]+' c 0,0 '+gsap.utils.wrap(pts,[i-nPts/2])+' '+gsap.utils.wrap(pts,[i-3]),
                'stroke-dasharray': lineLength + ' ' + lineLength,
                'stroke-dashoffset': lineLength
            },
        });

        stage.appendChild(p);

        const lineTL = gsap.timeline({
                repeat: -1,
                defaults: {
                    duration: 0.4
                }
            })
            .to(p, {
                attr: {
                    'stroke-dashoffset': -lineLength
                },
                ease: 'expo.inOut',
                duration: 0.6
            }, 0.4)

        if (i < nPts) {
            lineTL.to('.c' + i, {
                scale: 1.2,
                ease: 'expo.inOut'
            }, 0.1)
            lineTL.to('.c' + i, {
                scale: 0.8,
                ease: 'expo.in'
            }, 0.5)
        }

        tl.add(lineTL, 1 - i / nPts);
    }

    tl.play(5).timeScale(timeScale)
</script>
<script>
    $(function () {
        // loadingShow()
    })

    function loadingShow() {
        $("#loading").show();
    }

    function loadingHide() {
        $("#loading").hide();
    }
</script>

    <!-- <script src="https://cdn.bootcdn.net/ajax/libs/jquery.scrollbar/0.2.9/jquery.scrollbar.min.js"></script> -->
    <script>
        // $(document).ready(function () {
        //     $('.l_body').scrollbar({
        //         barWidth: 10, //滚动条的宽度(这里根据需要写数值即可，不设置是10,即默认10px)
        //         position: "x,y", //写“x”代表只出水平滚动条，写“y”表示只出垂直滚动条，写“x,y”则出水平和垂直滚动条（只有在内容超出容器时才出现滚动条）
        //         wheelDis: 15 //滚轮滚动一次向下或向上滚动的距离，默认是15，可根据需要修改数值
        //     });
        // });
    </script>
    <script>
        $('img').lazyload({
            threshold: 0,
            failure_limit: 0,
            event: "scroll", //触发事件
            effect: "show", //显示方式
            container: window, //容器
            data_attribute: "original", //属性
            skip_invisible: true,
            appear: null,
            load: null,
            placeholder: ""
        });
    </script>
<script>
            window.imageLazyLoadSetting = {
                isSPA: false,
                preloadRatio: 3,
                processImages: null,
            };
        </script><script>window.addEventListener("load",function(){var t=/\.(gif|jpg|jpeg|tiff|png)$/i,r=/^data:image\/[a-z]+;base64,/;Array.prototype.slice.call(document.querySelectorAll("img[data-original]")).forEach(function(a){var e=a.parentNode;"A"===e.tagName&&(e.href.match(t)||e.href.match(r))&&(e.href=a.dataset.original)})});</script><script>!function(n){n.imageLazyLoadSetting.processImages=o;var e=n.imageLazyLoadSetting.isSPA,i=n.imageLazyLoadSetting.preloadRatio||1,r=Array.prototype.slice.call(document.querySelectorAll("img[data-original]"));function o(){e&&(r=Array.prototype.slice.call(document.querySelectorAll("img[data-original]")));for(var t,a=0;a<r.length;a++)0<=(t=(t=r[a]).getBoundingClientRect()).bottom&&0<=t.left&&t.top<=(n.innerHeight*i||document.documentElement.clientHeight*i)&&function(){var t,e,n,i,o=r[a];t=o,e=function(){r=r.filter(function(t){return o!==t})},n=new Image,i=t.getAttribute("data-original"),n.onload=function(){t.src=i,e&&e()},t.src!==i&&(n.src=i)}()}o(),n.addEventListener("scroll",function(){var t,e;t=o,e=n,clearTimeout(t.tId),t.tId=setTimeout(function(){t.call(e)},500)})}(this);</script></body>